Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations

Pages: 1367 - 1391, Volume 29, Issue 4, April 2011      doi:10.3934/dcds.2011.29.1367

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Andrea L. Bertozzi - Department of Mathematics, UCLA, Los Angeles, CA, 90095, United States (email)
Ning Ju - Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK 74078, United States (email)
Hsiang-Wei Lu - Department of Mechanical Engineering, University of California, Los Angeles, CA, 90095-1555, United States (email)

Abstract: We consider a class of splitting schemes for fourth order nonlinear diffusion equations. Standard backward-time differencing requires the solution of a higher order elliptic problem, which can be both computationally expensive and work-intensive to code, in higher space dimensions. Recent papers in the literature provide computational evidence that a biharmonic-modified, forward time-stepping method, can provide good results for these problems. We provide a theoretical explanation of the results. For a basic nonlinear 'thin film' type equation we prove $H^1$ stability of the method given very simple boundedness constraints of the numerical solution. For a more general class of long-wave unstable problems, we prove stability and convergence, using only constraints on the smooth solution. Computational examples include both the model of 'thin film' type problems and a quantitative model for electrowetting in a Hele-Shaw cell (Lu et al J. Fluid Mech. 2007). The methods considered here are related to 'convexity splitting' methods for gradient flows with nonconvex energies.

Keywords:  Fourth order equations, convex splitting.
Mathematics Subject Classification:  65M06, 65M12, 65M70.

Received: December 2009;      Revised: July 2010;      Available Online: December 2010.